Topics in the Theory of Algebraic Groups

This article is a collection of notes from a series of talks given at the Bernoulli center. The attendees ranged from people who have never studied algebraic groups to experts. Consequently the series began with two introductory talks on the structure of algebraic groups, supplemented by two lectures of Steve Donkin on representation theory. The notes from his lectures appear in this volume and we encourage readers new to the theory of algebraic groups to consult his notes as this material will be used throughout this article. The first section of this article contains a brief introduction to the theory of algebraic groups starting from the classification of simple Lie algebras over C. The next section covers morphisms of simple algebraic groups and this leads to a discussion of the finite groups of Lie type and Lang’s theorem. Sections three and four concern maximal subgroups of algebraic groups. Section 5 concerns a problem on the finiteness of double coset spaces. The last two sections cover some current work on unipotent classes in simple algebraic groups. None of the sections represents anything near a complete account of the subject at hand so the interested reader should consult the references for further information.